Shadowing and hyperbolicity for linear delay difference equations

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this paper, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a)~for nonautonomous equations with finite delays and uniformly bounded compact coefficient operators in (possibly infinite-dimensional) Banach spaces, (b)~for Volterra difference equations with infinite delay in finite dimensional spaces.